A note on spherical maxima sharing the same Lagrange multiplier

In this paper, we establish a general result on spherical maxima sharing the same Lagrange multiplier of which the following is a particular consequence: Let $X$ be a real Hilbert space. For each $r>0$, let $S_r=\{x\in X : \|x\|^2=r\}$. Let $J:X\to {\bf R}$ be a sequentially weakly upper semicontinuous functional which is G\^ateaux differentiable in $X\setminus \{0\}$. Assume that $$\limsup_{x\to 0}{{J(x)}\over {\|x\|^2}}=+\infty\ .$$ Then, for each $\rho>0$, there exists an open interval $I\subseteq ]0,+\infty[$ and an increasing function $\varphi:I\to ]0,\rho[$ such that, for each $\lambda\in I$, one has $$\emptyset\neq \left \{x\in S_{\varphi(\lambda)} : J(x)=\sup_{S_{\varphi(\lambda)}}J\right\}\subseteq \{x\in X : x=\lambda J'(x)\}\ .$$